of Schwartz, distributions that are more singular than the Dirac delta function are always negative somewhere.) Such "negative probability" or high degree of singularity is a feature inherent to the representation and does not diminish the meaningfulness of expectation values taken with respect to . Even if does behave like an ordinary probability distribution, however, the matter is not quite so simple. According to Mandel and Wolf: "The different coherent states are not [mutually] orthogonal, so that even if behaved like a true probability density [function], it would not describe probabilities of mutually exclusive states." Examples Thermal radiation From statistical mechanics arguments in the Fock basis, the mean photon number of a mode with wavevector and polarization state for a black body at temperature is known to be The representation of the black body is In other words, every mode of the black body is normally distributed in the basis of coherent states. Since is positive and bounded, this system is essentially classical. This is actually quite a remarkable result because for thermal equilibrium the density matrix is also diagonal in the Fock basis, but Fock states are non-classical. Highly singular example Even very simple-looking states may exhibit highly non-classical behavior. Consider a superposition of two coherent states where are constants subject to the normalizing constraint Note that this is quite different from a qubit because and are not orthogonal. As it is straightforward to calculate , we can use the Mehta formula above to compute , Despite having infinitely many derivatives of delta functions, still obeys the optical equivalence theorem. If the expectation value of the number operator, for example, is taken with respect to the state vector or as a phase space average with respect to , the two expectation values match: See also Quasiprobability distribution#Characteristic functions Nonclassical light Wigner quasiprobability distribution Husimi Q representation Nobel Prize controversies References Citations Bibliography Quantum optics